Fake Projective Planes
نویسنده
چکیده
1.1. A fake projective plane is a smooth compact complex surface which is not the complex projective plane but has the same Betti numbers as the complex projective plane. Such a surface is known to be projective algebraic and it is the quotient of the (open) unit ball B in C (B is the symmetric space of PU(2, 1)) by a torsion-free cocompact discrete subgroup of PU(2, 1) whose Euler-Poincaré characteristic is 3. These surfaces have the smallest EulerPoincaré characteristic among all smooth surfaces of general type. The first fake projective plane was constructed by David Mumford [Mu] using p-adic uniformization, and later two more examples were found by M. Ishida and F. Kato in [IK] using a similar method. In [Ke] JongHae Keum [Ke] has constructed an example which is birational to a cyclic cover of degree 7 of a Dolgachev surface (see 5.15 below). It is known that there are only finitely many fake projective planes ([Mu]), and an important problem in complex algebraic geometry is to determine them all.
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